In this chapter, we provide a guided tour through our theory of combinatorial connection matrices, supplemented by numerous examples. This includes a summary of our definition of connection matrices, as well as outlines of the main results of the book. We also introduce the necessary background material, albeit not in full technical detail. In this way, the current chapter serves as a quick introduction to the theory, while avoiding some of the more technical aspects. Precise definitions, statements, and proofs will be given in the sequel. In addition, we briefly address the computability of connection matrices and demonstrate how in combination with the theory of multivector fields they can be used to analyze the global dynamics of low-dimensional ordinary differential equations.